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Difference between revisions of "Mock AIME 1 2007-2008 Problems"

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== Problem 1 ==
 
== Problem 1 ==
 
What is the coefficient of <math>x^3y^{13}</math> in <math>\left(\frac 12x + y\right)^{17}</math>?
 
What is the coefficient of <math>x^3y^{13}</math> in <math>\left(\frac 12x + y\right)^{17}</math>?
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 
The expansion of <math>(x+1)^n</math> has 3 consecutive terms with coefficients in the ratio <math>1:2:3</math> that can be written in the form <cmath>{n\choose k} : {n\choose k+1} : {n \choose k+2}</cmath>
 
The expansion of <math>(x+1)^n</math> has 3 consecutive terms with coefficients in the ratio <math>1:2:3</math> that can be written in the form <cmath>{n\choose k} : {n\choose k+1} : {n \choose k+2}</cmath>
 
Find the sum of all possible values of <math>n+k</math>.  
 
Find the sum of all possible values of <math>n+k</math>.  
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 
A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?
 
A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 
If <math>x</math> is an odd number, then find the largest integer that always divides the expression
 
If <math>x</math> is an odd number, then find the largest integer that always divides the expression
 
<cmath>(10x+2)(10x+6)(5x+5)</cmath>  
 
<cmath>(10x+2)(10x+6)(5x+5)</cmath>  
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 
Let <math>S = (1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.
 
Let <math>S = (1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
−
[problem to be filled in]
+
A <math>\frac 1p</math> -array is a structured, infinite, collection of numbers. For example, a <math>\frac 13</math> -array is constructed as follows:
 +
 +
<cmath>\begin{align*}
 +
1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\
 +
\frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\
 +
\frac 1{36} \qquad \frac 1{108} \qquad &\cdots\\
 +
\frac 1{216} \qquad &\cdots\\
 +
&\ddots
 +
\end{align*}</cmath>
 +
 +
In general, the first entry of each row is <math>\frac{1}{2p}</math> times the first entry of the previous row. Then, each succeeding term in a row is <math>\frac 1p</math> times the previous term in the same row. If the sum of all the terms in a <math>\frac{1}{2008}</math> -array can be written in the form <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find the remainder when <math>m+n</math> is divided by <math>2008</math>.
 +
 
 +
[[Mock AIME 1 2007-2008 Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
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<cmath>(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1</cmath>
 
<cmath>(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1</cmath>
 
Find <math>g(2)</math>.
 
Find <math>g(2)</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 
A sequence of ten <math>0</math>s and/or <math>1</math>s is randomly generated. If the probability that the sequence does not contain two consecutive <math>1</math>s can be written in the form <math>\dfrac{m}{n}</math>, where <math>m,n</math> are relatively prime positive integers, find <math>m+n</math>.
 
A sequence of ten <math>0</math>s and/or <math>1</math>s is randomly generated. If the probability that the sequence does not contain two consecutive <math>1</math>s can be written in the form <math>\dfrac{m}{n}</math>, where <math>m,n</math> are relatively prime positive integers, find <math>m+n</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
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: <math>\frac n2</math> is a perfect square.  
 
: <math>\frac n2</math> is a perfect square.  
 +
 
: <math>\frac n3</math> is a perfect cube.  
 
: <math>\frac n3</math> is a perfect cube.  
 +
 
: <math>\frac n5</math> is a perfect fifth.
 
: <math>\frac n5</math> is a perfect fifth.
  
 
How many divisors does <math>n</math> have that are not multiples of 10?
 
How many divisors does <math>n</math> have that are not multiples of 10?
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
An oreo shop sells <math>5</math> different flavors of oreos and <math>3</math> different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).
+
An oreo shop sells <math>5</math> different flavors of oreos and <math>3</math> different flavors of milk. Alpha and Beta decide to purchase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).
 +
 
 +
[[Mock AIME 1 2007-2008 Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
<math>\triangle DEF</math> is inscribed inside <math>\triangle ABC</math> such that <math>D,E,F</math> lie on <math>BC, AC, AB</math>, respectively. The circumcircles of <math>\triangle DEC, \triangle BFD, \triangle AFE</math> have centers <math>O_1,O_2,O_3</math>, respectively. Also, <math>AB = 23, BC = 25, AC=24</math>, and <math>\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}</math>. The length of <math>BD</math> can be written in the form <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 
Let <math>d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2}</math> and <math>d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2}</math>. If <math>1 \le a \le 251</math>, how many integral values of <math>a</math> are there such that <math>d_1 \cdot d_2</math> is a multiple of <math>5</math>?
 
Let <math>d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2}</math> and <math>d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2}</math>. If <math>1 \le a \le 251</math>, how many integral values of <math>a</math> are there such that <math>d_1 \cdot d_2</math> is a multiple of <math>5</math>?
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
Line 48: Line 85:
 
<cmath>\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}</cmath>
 
<cmath>\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}</cmath>
 
for <math>x \in \mathbb{R}</math> such that both sides are defined. Find <math>F(12)</math>.
 
for <math>x \in \mathbb{R}</math> such that both sides are defined. Find <math>F(12)</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 
Points <math>A</math> and <math>B</math> lie on <math>\odot O</math>, with radius <math>r</math>, so that <math>\angle OAB</math> is acute. Extend <math>AB</math> to point <math>C</math> so that <math>AB = BC</math>. Let <math>D</math> be the intersection of <math>\odot O</math> and <math>OC</math> such that <math>CD = \frac {1}{18}</math> and <math>\cos(2\angle OAC) = \frac 38</math>. If <math>r</math> can be written as <math>\frac{a+b\sqrt{c}}{d}</math>, where <math>a,b,</math> and <math>d</math> are relatively prime and <math>c</math> is not divisible by the square of any prime, find <math>a+b+c+d</math>.
 
Points <math>A</math> and <math>B</math> lie on <math>\odot O</math>, with radius <math>r</math>, so that <math>\angle OAB</math> is acute. Extend <math>AB</math> to point <math>C</math> so that <math>AB = BC</math>. Let <math>D</math> be the intersection of <math>\odot O</math> and <math>OC</math> such that <math>CD = \frac {1}{18}</math> and <math>\cos(2\angle OAC) = \frac 38</math>. If <math>r</math> can be written as <math>\frac{a+b\sqrt{c}}{d}</math>, where <math>a,b,</math> and <math>d</math> are relatively prime and <math>c</math> is not divisible by the square of any prime, find <math>a+b+c+d</math>.
 +
 +
[[Mock AIME 1 2007-2008 Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 
The sum
 
The sum
 
<cmath>\sum_{x=2}^{44} 2\sin{x}\sin{1}[1 + \sec (x-1) \sec (x+1)]</cmath>
 
<cmath>\sum_{x=2}^{44} 2\sin{x}\sin{1}[1 + \sec (x-1) \sec (x+1)]</cmath>
−
can be written in the form <math>\sum_{n=1}^{4} (-1)^n \frac{\Phi(\theta_n)}{\Psi(\theta_n)}, where </math>\Phi,\, \Psi<math></math> are trigonometric functions and <math>\theta_1,\, \theta_2, \, \theta_3, \, \theta_4</math> are degrees <math>\in [0,45]</math>. Find <math>\theta_1 + \theta_2 + \theta_3 + \theta_4</math>.
+
can be written in the form <math>\sum_{n=1}^{4} (-1)^n \frac{\Phi(\theta_n)}{\Psi(\theta_n)}</math>, where <math>\Phi,\, \Psi</math> are trigonometric functions and <math>\theta_1,\, \theta_2, \, \theta_3, \, \theta_4</math> are degrees <math>\in [0,45]</math>. Find <math>\theta_1 + \theta_2 + \theta_3 + \theta_4</math>.
 +
 
 +
[[Mock AIME 1 2007-2008 Problems/Problem 15|Solution]]

Latest revision as of 14:15, 13 February 2018

Problem 1

What is the coefficient of $x^3y^{13}$ in $\left(\frac 12x + y\right)^{17}$?

Solution

Problem 2

The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\] Find the sum of all possible values of $n+k$.

Solution

Problem 3

A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?

Solution

Problem 4

If $x$ is an odd number, then find the largest integer that always divides the expression \[(10x+2)(10x+6)(5x+5)\]

Solution

Problem 5

Let $S = (1+i)^{17} - (1-i)^{17}$, where $i=\sqrt{-1}$. Find $|S|$.

Solution

Problem 6

A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:

\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108} \qquad &\cdots\\ \frac 1{216} \qquad &\cdots\\ &\ddots \end{align*}

In general, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row. Then, each succeeding term in a row is $\frac 1p$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{2008}$ -array can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $2008$.

Solution

Problem 7

Consider the following function $g(x)$ defined as \[(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1\] Find $g(2)$.

Solution

Problem 8

A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.

Solution

Problem 9

Let $n$ represent the smallest integer that satisfies the following conditions:

$\frac n2$ is a perfect square.
$\frac n3$ is a perfect cube.
$\frac n5$ is a perfect fifth.

How many divisors does $n$ have that are not multiples of 10?

Solution

Problem 10

An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purchase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways could they have left the store with 3 products collectively? (A possible purchase is Alpha purchases 1 box of uh-oh oreos and 1 gallon of whole milk while Beta purchases 1 bag of strawberry milkshake oreos).

Solution

Problem 11

$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}$. The length of $BD$ can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

Solution

Problem 12

Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2}$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2}$. If $1 \le a \le 251$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $5$?

Solution

Problem 13

Let $F(x)$ be a polynomial such that $F(6) = 15$ and \[\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}\] for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$.

Solution

Problem 14

Points $A$ and $B$ lie on $\odot O$, with radius $r$, so that $\angle OAB$ is acute. Extend $AB$ to point $C$ so that $AB = BC$. Let $D$ be the intersection of $\odot O$ and $OC$ such that $CD = \frac {1}{18}$ and $\cos(2\angle OAC) = \frac 38$. If $r$ can be written as $\frac{a+b\sqrt{c}}{d}$, where $a,b,$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime, find $a+b+c+d$.

Solution

Problem 15

The sum \[\sum_{x=2}^{44} 2\sin{x}\sin{1}[1 + \sec (x-1) \sec (x+1)]\] can be written in the form $\sum_{n=1}^{4} (-1)^n \frac{\Phi(\theta_n)}{\Psi(\theta_n)}$, where $\Phi,\, \Psi$ are trigonometric functions and $\theta_1,\, \theta_2, \, \theta_3, \, \theta_4$ are degrees $\in [0,45]$. Find $\theta_1 + \theta_2 + \theta_3 + \theta_4$.

Solution

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